Medium ACT Word Problems Practice Questions

Mastering Medium ACT Word Problems requires translating complex English scenarios into precise mathematical equations and solving them efficiently under time pressure. These problems often sit in the middle of the math section, bridging the gap between basic arithmetic and advanced trigonometry. To succeed, students must identify the core mathematical concept hidden within a narrative, whether it involves algebra, geometry, or ACT Statistics. By practicing these intermediate-level challenges, you can build the stamina and analytical skills necessary to reach your target score on the ACT Prep journey.

Concept Explanation

Medium ACT word problems are mathematical tasks that require multiple steps to solve, typically involving the translation of a real-world scenario into algebraic expressions or geometric models. Unlike easy questions that might ask for a single calculation, medium-level problems often combine two or more concepts, such as finding a percentage and then applying it to a ratio. The key to solving these is a systematic approach: identifying the unknown variable, extracting relevant constants from the text, and setting up an equation. Many of these problems also test your ability to work with units, such as converting minutes to hours or feet to yards, which is a common staple of ACT Math Practice. Understanding how to navigate these linguistic traps is essential for any student looking to excel on the exam, as noted by educational resources like Khan Academy.

Solved Examples

1. Example 1: Linear Equation Modeling
   A car rental agency charges a flat daily fee of $25 plus $0.15 per mile driven. If Sarah rented a car for one day and her total bill was $46.60, how many miles did she drive?
   1. Identify the components of the total cost: Total = Flat Fee + (Rate $\times$ Miles).
   2. Set up the equation: $$46.60 = 25 + 0.15m$$
   3. Subtract the flat fee from both sides: $$21.60 = 0.15m$$
   4. Divide by 0.15 to solve for $m$: $$m = \frac{21.60}{0.15} = 144$$
   5. Sarah drove 144 miles.
2. Example 2: Percentage and Algebra
   A jacket is on sale for 20% off the original price. After the discount is applied, a 5% sales tax is added to the sale price. If the final amount paid is $100.80, what was the original price of the jacket?
   1. Let $x$ be the original price.
   2. The sale price is $0.80x$.
   3. The price with tax is $1.05(0.80x)$.
   4. Set up the equation: $$1.05(0.80x) = 100.80$$
   5. Simplify the left side: $$0.84x = 100.80$$
   6. Divide by 0.84: $$x = \frac{100.80}{0.84} = 120$$
   7. The original price was $120.
3. Example 3: Average and Sums
   The average of five test scores is 82. If the first four scores are 78, 85, 90, and 75, what must the fifth score be to maintain the average?
   1. Calculate the total sum needed: $$82 \times 5 = 410$$
   2. Sum the known scores: $$78 + 85 + 90 + 75 = 328$$
   3. Subtract the known sum from the total sum: $$410 - 328 = 82$$
   4. The fifth score must be 82.

Practice Questions

1. A rectangular garden has a perimeter of 50 feet. If the length is 5 feet longer than the width, what is the area of the garden in square feet?
2. A water tank is $\frac{1}{4}$ full. After adding 12 gallons of water, the tank is $\frac{2}{3}$ full. What is the total capacity of the tank in gallons?
3. A group of friends goes out for dinner. The total bill is $150. They want to leave a 18% tip on the total bill and then split the entire amount (bill + tip) equally among 4 people. How much does each person owe?

4. A cyclist travels at an average speed of 12 miles per hour for 45 minutes and then at 18 miles per hour for 30 minutes. What is the total distance traveled?
5. A store sells red pens in packs of 6 and blue pens in packs of 8. If a teacher wants to buy the same number of red and blue pens, what is the minimum total number of pens they must buy?
6. A photographer charges a sitting fee of $50 and $15 per printed photo. If a customer spends exactly $155, how many photos did they purchase?
7. The sum of three consecutive even integers is 72. What is the value of the largest integer?
8. An aquarium is 24 inches long, 10 inches wide, and 12 inches high. If it is filled with water to a height of 9 inches, what is the volume of the water in cubic inches?
9. In a class of 30 students, the ratio of boys to girls is 2:3. If 2 more boys join the class, what is the new ratio of boys to girls?
10. A machine can produce 400 widgets in 5 hours. At this rate, how many hours would it take to produce 1,000 widgets?

Answers & Explanations

1. Answer: 150
   Let width be $w$ and length be $w+5$. Perimeter $P = 2(l + w)$. So, $$50 = 2(w + 5 + w) \rightarrow 50 = 2(2w + 5) \rightarrow 25 = 2w + 5 \rightarrow 20 = 2w \rightarrow w = 10$$. If $w = 10$, then $l = 15$. Area = $10 \times 15 = 150$.
2. Answer: 28.8
   Let $C$ be the capacity. $$\frac{1}{4}C + 12 = \frac{2}{3}C$$. Subtracting $\frac{1}{4}C$ from both sides: $$12 = \frac{2}{3}C - \frac{1}{4}C$$. Find a common denominator (12): $$12 = \frac{8}{12}C - \frac{3}{12}C \rightarrow 12 = \frac{5}{12}C$$. Multiply by $\frac{12}{5}$: $$C = 12 \times \frac{12}{5} = \frac{144}{5} = 28.8$$.
3. Answer: $44.25
   Total with tip = $$150 \times 1.18 = 177$$. Split by 4: $$177 / 4 = 44.25$$.
4. Answer: 18 miles
   For the first leg: $12 \text{ mph} \times 0.75 \text{ hours} (45/60) = 9 \text{ miles}$. For the second leg: $18 \text{ mph} \times 0.5 \text{ hours} (30/60) = 9 \text{ miles}$. Total = $9 + 9 = 18$.
5. Answer: 48
   Find the Least Common Multiple (LCM) of 6 and 8. Multiples of 6: 6, 12, 18, 24... Multiples of 8: 8, 16, 24... The LCM is 24. Since they need 24 of each, the total pens = $24 + 24 = 48$.
6. Answer: 7
   Equation: $$50 + 15p = 155 \rightarrow 15p = 105 \rightarrow p = 7$$.
7. Answer: 26
   Let the integers be $n$, $n+2$, and $n+4$. $$n + (n+2) + (n+4) = 72 \rightarrow 3n + 6 = 72 \rightarrow 3n = 66 \rightarrow n = 22$$. The largest is $22 + 4 = 26$.
8. Answer: 2,160
   Volume = $length \times width \times height$. Use the water height, not the tank height: $$24 \times 10 \times 9 = 2,160$$.
9. Answer: 7:9
   Total parts = $2 + 3 = 5$. One part = $30 / 5 = 6$. Boys = $2 \times 6 = 12$, Girls = $3 \times 6 = 18$. New boys = $12 + 2 = 14$. New ratio = 14:18, which simplifies to 7:9.
10. Answer: 12.5
    Rate = $400 / 5 = 80$ widgets per hour. Time = $1,000 / 80 = 12.5$ hours.

1. A movie theater sells adult tickets for $12 and child tickets for $8. If a group buys 10 tickets for a total of $96, how many adult tickets were purchased?

• A3
• B4
• C5
• D6

Frequently Asked Questions

How do I translate "at most" or "at least" in ACT word problems?

In the context of the ACT, "at most" corresponds to the less than or equal to symbol ($\leq$), while "at least" corresponds to the greater than or equal to symbol ($\geq$). These are frequently used in ACT Algebra inequality problems where you must define a range of possible values.

What is the most common mistake in multi-step word problems?

The most common error is forgetting to perform the final step or failing to convert units, such as using minutes in an equation where the rate is given in hours. Using a tool like the AI Question Generator can help you practice identifying these subtle traps in varied scenarios.

Do I need to memorize specific formulas for word problems?

Yes, you should be comfortable with distance ($d = rt$), interest ($I = Prt$), and basic area/volume formulas. Many medium-level problems require applying these to scenarios involving ACT Geometry or motion.

How can I get faster at reading word problems?

Focus on "math-sketching" by underlining numbers and identifying the question's goal (e.g., "find the width") before you start calculating. Regular use of an AI Exam Simulator can improve your reading speed and comprehension under timed conditions.

Are there word problems on the ACT that involve probability?

Yes, word problems frequently incorporate ACT Probability by describing a selection process from a group. You will often need to calculate the total number of outcomes first based on a narrative description.